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A327645
Number of proper n-times partitions of 2n.
2
1, 1, 6, 88, 2489, 112669, 8204101, 799422247, 109633217402, 19157475773052, 4260985739868007, 1161511740640164091, 388990633971649889649, 152369510393132343133762, 70914541309488196549283707, 38152280583855500772704976704, 23639325145221113389859164367779
OFFSET
0,3
COMMENTS
In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.
LINKS
FORMULA
a(n) = A327639(2n,n).
EXAMPLE
a(2) = 6: 4->31->211, 4->31->1111, 4->22->112, 4->22->211, 4->22->1111, 4->211->1111.
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
end:
a:= n-> add(b(2*n$2, i)*(-1)^(n-i)*binomial(n, i), i=0..n):
seq(a(n), n=0..17);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
a[n_] := Sum[b[2n, 2n, i] (-1)^(n - i) Binomial[n, i], {i, 0, n}];
a /@ Range[0, 17] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)
CROSSREFS
Cf. A327639.
Sequence in context: A177567 A177563 A177569 * A309165 A127183 A054952
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 20 2019
STATUS
approved